How many person sits between A and L, when counted left

Solution

We have circumference of the circular table = 306m. Person sits at consecutive multiple of 3. Thus, (3 + 6 + 9 + 12 + 15 + …….. + 3N) = 306 Similarly, if we take 3 common from both side, we have: (1 + 2 + 3 + 4 + 5 + …… + N) = 102 We know that, sum of N consecutive numbers = N (N + 1)/2 Thus, N (N + 1)/2 = 102 We have: Case 1: If N = 14, then (14 x 15)/2 = 105, which is more than the given value. Case 2: If N = 13, then (13 x 14)/2 = 91, which is less than the given value. That means total number of person in the circle are either 13 or less than 13. Again, for 13 number of person: Circumference = (3 + 6 + 9 + 12 + …………. + 39) = 273, which is less than the given circumference. Thus, we have possibility that counting starts from 6, then we have: Circumference = (6 + 9 + 12 + 15 + ………………. + 42) = 312. Which is more than given circumference, thus there must be less 13 persons. Again, for 12 person Circumference = (3 + 6 + 9 + 12 + …………. + 36) = 234, which is less than the given circumference. Thus, we have possibility that counting starts from 6, then we have: Circumference = (6 + 9 + 12 + 15 + ………………. + 39) = 270, which is still less than the given circumference, Thus, we have possibility that counting starts from 9, then we have: Circumference = (9 + 12 + 15 + ………………. + 42) = 306 That, means we have 12 person in the circle and counting starts from 9 in clockwise direction.  N sits 39m right of Q, who doesn’t sits adjacent to O but sits facing center, that means we have two possible combination for distance to be 39m, i.e. (18 + 21) & 39, that means in case (1) one person sits between N and Q, in case (2) no person sits between N and Q. Three person sits between X and N, who neither sits adjacent to G nor sits adjacent to W. X sits immediate left of O. Two person sits between O and E. E sits 51m left of G, only two such combinations are possible (24, 27) & (42, 9), that means in case (1) O sits facing away from center and E sits (42 + 9)q left of G and case (2) is not valid. Based on above given information we have: Case (2) is not valid as E sits 51m left of G. Again, we have: A sits immediate left of E and is four places away from W. W neither sits adjacent to G nor sits adjacent to L, that means in case (1a) E sits facing away from center and W sits adjacent to X, in case (1b) E sits facing center and W sits adjacent to N. X sits third to right of L.L and A sits facing in same direction, that means in case (1a) L sits facing away from center, in case (1b) L sits facing towards center. F sits 21m right of J. F neither sits adjacent to N nor sits facing same direction with Q, since only such possible combination are (9 + 12)q & 21m, that means in case (1a) J sits facing center and F sits facing away from center and case (1b) is not valid. Not more than four person sits between L and H, when counted left of That means H sits facing away from center and sits immediate right of O. X and N sits facing in same direction but opposite to H, since H sits facing away from center, that means X and N sits facing center. Based on above given information we have final arrangement as follow: Case (1b) is not valid as F sits 21m right of J.